Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperature, T1 and T2 . The temperature of the hot reservoir of the first engine is T1 and the temperature of the cold reservoir of the second engine is T2 . T is temperature of the sink of first engine which is also the source for the second which is also the source for the second engine. How is T related to T1 and T2 . If both engines perform equal amount of work?
Solution
Q<sub>H</sub> : Heat input to 1<sup>st</sup> engine
<br><br>Q<sub>L</sub>
: Heat rejected from 1<sup>st</sup> engine
<br><br>Q'<sub>L</sub> : Heat rejected from 2<sup>nd</sup> engine
<br><br>T : Lower temperature of first engine
<br><br>Work done by 1<sup>st</sup> engine = work done by 2<sup>nd</sup> engine
<br><br>Q<sub>H</sub>
– Q<sub>L</sub> = Q<sub>L</sub>
– Q'<sub>L</sub>
<br><br>$\Rightarrow$ 2Q<sub>L</sub>
= Q<sub>H</sub>
+ Q'<sub>L</sub>
<br><br>$\Rightarrow$ 2 = ${{{Q_H}} \over {{Q_L}}} + {{Q{'_L}} \over {{Q_L}}}$
<br><br>$\Rightarrow$ 2 = ${{{T_1}} \over T} + {{{T_2}} \over T}$
<br><br>$\Rightarrow$ $T = {{{T_1} + {T_2}} \over 2}$
About this question
Subject: Physics · Chapter: Thermodynamics · Topic: Zeroth and First Law
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